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DEARER: A Distance-and-Energy-Aware Routing with
Energy Reservation for Energy Harvesting Wireless
Sensor Networks
Yunquan Dong†, Jian Wang†, Byonghyo Shim†, Senior Member, IEEE,
and Dong In Kim‡, Senior Member, IEEE†Dept. of Electrical & Computer Engineering, Seoul National University, Korea
‡School of Information & Communication Engineering, Sungkyunkwan
University, Korea
[email protected], [email protected], [email protected], [email protected]
AbstractIn an energy harvesting wireless sensor network (EH-WSN), an energy-harvesting unit is used at
each sensor node to collect energy from ambient environments. This ensures a perpetual energy supply
to the network. However, due to limitation of energy intensity and randomness of energy arrivals, there
is a mismatch between the real-time energy supply and the actual demand at each sensor node, resulting
in irregular and sporadic energy shortage events. With an aim to combat the energy shortage issue in EH-
WSNs, we develop a cluster-based routing protocol, referred to as the distance-and-energy-aware routing
with energy reservation (DEARER). In DEARER, all nodes take turns to serve as cluster head (CH)
nodes to forward packets, which are collected from non-CH nodes, to the sink. In particular, those nodes
with large energy arrival rate or being closer to the sink are endowed with high chance to serve as the
CH nodes. When a node works in the non-CH mode (i.e., without relaying), it reserves a portion of the
harvested energy for future use in the CH mode. The proposed strategy cuts down energy-consuming
transmissions from nodes to the sink while ensuring nodes more energy for the CH mode, thereby
reducing energy shortage events at the CH nodes. By theoretical analysis and numerical simulations,
we study the performance of DEARER in terms of the network energy efficiency, transmission ratio,
and CH-outage probability. Our results demonstrate that the proposed DEARER protocol outperforms
direct transmission and also approaches the genie-aided routing where CH nodes are selected based on
real-time energy information of nodes.
Index Terms
Routing, energy harvesting, energy reservation, energy efficiency, outage probability.
DRAFT
2
I. INTRODUCTION
Wireless sensor network (WSN) has been widely deployed for various purposes, such as remote
environment monitoring [1], health-care [2], and air quality monitoring [3]. By employing a
large number of low-power sensors, a WSN can collect desired information in a distributed and
self-organized manner. Such feature has made the WSN a key enabler to realize the Internet-of-
Things (IoT) and green communication era [4], [5]. One well-known concern in WSNs is that the
network lifetime is greatly restrained by the battery capacity of sensor nodes [1]–[3]. To address
this issue, many efforts have been made in recent years, focusing on either exploiting sustainable
energy to diversify the energy supply or greening network to reduce energy expenditure. Among
these, energy harvesting has received considerable attention owing to its ability in extending
the lifetime of WSNs [6]. By utilizing an energy harvesting unit and an energy buffer, each
sensor node of a WSN can harvest energy (e.g., solar and wind power) from environments,
thereby ensuring an unlimited energy supply to each node. This type of network is referred to
as the energy harvesting WSN (EH-WSN) [6]. Sustainable energy supply techniques have also
been used in hyper cellular networks [7]. The applications can be further extended with wireless
energy transfer technologies [8].
As an alternative means to prolong the network lifetime of WSNs, greening network with
energy efficient routing methods has been studied extensively [9]–[14]. The methods roughly
fall into two categories: approaches relying on flat routing protocols [9], [10] and schemes
based on hierarchical routing protocols [11], [12]. While each node plays the same role in the
flat protocols, some nodes in the hierarchical protocols are used as cluster head (CH) nodes to
forward packets of non-CH nodes to the sink. In general, approaches based on the hierarchical
protocols outperform those using the flat protocols. Also, by incorporating more information of
the network (e.g., geographic information and real-time energy status), further improvement of
performance can be achieved [13], [14]. Such protocols, typically employing a central controller,
are particularly preferable in the EH-WSN scenarios since the central controller can optimize
routing paths by using the real-time energy status of nodes [15], [16]. In addition, protocols
without a central controller yet achieving the centralized performance have also been proposed.
For example, the DEHAR protocol identifies the most energy-efficient route with the aid of real-
time node-to-node energy status exchanges [19]. However, due to extensive message exchanges,
DRAFT
3
these protocols often suffer from heavy signaling overhead [19] and also are inefficient in coping
with the randomness of energy arrivals in the EH-WSN, resulting in occasional energy shortage
events. To be specific, since the available ambient energy is susceptible to environmental factors
(e.g., light intensity for solar power and temperature for thermal power), the harvested energy
at each node is irregular and random, and hence may not meet the demand. In fact, since CH
nodes transmitting packets to the remote sink is more energy-consuming than non-CH nodes
transmitting packets to adjacent CH nodes, energy shortage often occur to those CH nodes.
The primary purpose of this paper is to propose a new routing protocol, referred to as the
distance-and-energy-aware routing with energy reservation (DEARER), to overcome the energy
shortage problem and also improve the energy efficiency of EH-WSNs. Our approach is inspired
by the observation that energy shortage often occurs at the CH nodes. In the DEARER paradigm,
the network operates in rounds. In a round, each node works either in the non-CH mode or in the
CH mode, where a node is said to be in the CH (non-CH) mode if it acts as a CH (non-CH) node.
When a node is in the non-CH mode, a portion of the harvested energy is stored as a “strategic
reserve” for future use. In this way, each node can reserve more energy for the upcoming
CH mode, thereby reducing the probability of energy shortage. Another important feature of
DEARER to reduce the energy shortage of CH nodes is to balance the energy consumption over
the network. To be specific, in each round, “enabler” nodes, which are closer to the sink (i.e.,
lower energy consumption in transmission) or have higher energy arrival rate (i.e., more resistant
to energy shortage), are endowed with more chance to serve as CH nodes. This essentially lowers
the average energy consumption and provides CH nodes with more energy, thus reducing the
probability of energy shortage. As a result, more packets can be delivered to the sink successfully,
and hence the overall energy efficiency of the network is improved.
The main contributions of this paper are as follows:
• We propose a cluster-based routing protocol called DEARER. By employing an energy-and-
distance-aware head selection algorithm and a proactive energy reservation policy, DEARER
reduces the energy shortage events at CH nodes and improves the network energy efficiency.
• We present a model for evaluating the performance of routing algorithms in EH-WSNs. In
the model, we use the network energy efficiency to characterize how many packets can be
successfully delivered to the sink for given amount of energy. We also parameterize the
transmission efficiency and reliability at each node in terms of the transmission ratio and
DRAFT
4
CH-outage probability. For these metrics, we provides explicit analytical expressions and
efficient numerical evaluations.
• We compare the performance of the DEARER protocol with direct transmission (i.e., the
worst strategy) and genie-aided routing (i.e., the best strategy). Our numerical and simulation
results demonstrate that DEARER outperforms direct transmission and approaches genie-
aided routing for most of realistic scenarios.
The rest part of the paper is organized as follows. In Section II, we present the network model,
multiple access model, and energy harvesting model. We describe the DEARER routing protocol
in Section III. The performance analysis of DEARER, including transmission ratio, CH-outage
probability, and network energy efficiency, are provided in Section IV. Numerical and simulation
results are presented in Section V. Finally, we conclude our work in Section VI.
II. SYSTEM MODEL
A. Network Model
We consider an EH-WSN with N sensor nodes and one sink node. The sensor nodes collect
information independently and send it to the sink according to a given routing protocol. Each
sensor node is equipped with an energy harvesting unit so that it can also collect energy (e.g.,
solar energy and wind energy) from ambient environments. The harvested energy is assumed
to be used only for the radio frequency (RF) unit. The sensor nodes are deployed uniformly
in a square area of 2L × 2L m2 (see Figure 1). Let (xn, yn) and (xs, ys) be the position of
an arbitrary node and the position of the sink, respectively, the distance between them is dn =√(xn − xs)2 + (yn − ys)2. For two nodes with position (xn1 , yn1) and (xn2 , yn2), the piecewise
distance is dn1n2 =√
(xn1 − xn2)2 + (yn1 − yn2)
2.
In this paper, we consider the cluster-based routing protocol for an energy hungry EH-WSN,
in which sensor nodes are classified into two categories: CH nodes and non-CH nodes. Each
non-CH node sends information to the nearest CH node, and the CH node forwards the collected
information to the sink. A CH node together with its non-CH members forms a cluster. In the
cluster-based routing, the network works in rounds and each round contains two phases: the
setup phase and the steady phase. In the setup phase, the CH nodes are selected and the clusters
are formed. In the steady phase, each non-CH node transmits the collected information to its
CH node. Upon receiving all the packets from its non-CH members, a CH node aggregates
DRAFT
5
(xs,ys)L
Sink
O
L
y
x
Non-CH node
CH node
Figure 1. The DEARER based network model.
the packets into one and forwards it to the sink. This type of hierarchical routing is beneficial
in various aspects: First, a large portion of direct transmissions to the sink, which are energy-
consuming, are prevented, and hence the quick energy drainage of nodes can be avoided. Second,
by aggregating information at CH nodes, the amount of information to be transmitted to the sink
can be reduced significantly. For example, a CH node may compress the information collected
from adjacent non-CH nodes with compressed sensing (CS) techniques [17], [18] and send only
the compression to the sink. Third, periodic re-assignment of CH nodes can balance the energy
consumption among nodes and thus improve the network lifetime. This is especially important
for EH-WSNs since sensors can ensure time to accumulate the needed energy. Finally, the
hierarchical routing strategy can be readily applied to large-scale networks.
B. Multiple Access Model
We assume that the time is slotted and each node communicates with its destination in a time
division multiple access (TDMA) manner.1 As shown in Figure 2, each frame consists of N slots
1Our results are readily extendable to other multiple access systems, such as the frequency division multiple access (FDMA)
and the code division multiple access (CDMA) systems, by using sub-band or codeword (instead of time slot) as resource blocks.
DRAFT
6
The network operates in rounds
lth
round:
i th frame:
Setup phase Steady phase
1 2 ... l ...
1 2 ... NF
jth
slot:
1 2 ... j ... N
1 2 Lp... ...
Figure 2. The frame structure under DEARER strategy.
and each of them is uniquely assigned to a node. In addition, the length of a slot is ts during
which a packet of Lp bits is transmitted. As mentioned, a round consists of a setup phase and a
steady phase. Although the setup phase is an overhead to the network operation, the portion of
this overhead is short compared to the steady phase that consists of NF frames of information
transmissions. After receiving all packets from its non-CH members, a CH node transmits the
aggregated packet to the sink as long as it has enough energy.
C. Energy Model and Channel Model
In this subsection, we describe the energy model of how sensor nodes harvest, store, and use
the energy. Before we proceed, we provide key assumptions used in our analysis.
i) The capacity of battery is assumed to be infinity. For example, the capacity of a small button
battery is more than 200 milliampere hour (mAh), which is large enough for most of energy
harvesting scenarios [21].
ii) Energy arrivals may occur at any time but the harvested energy can be used only from the
next frame.
iii) Packets received at each CH node are aggregated into one.
iv) The energy arrival at each node follows a stationary and ergodic Poisson process with
parameter λn [22]:
Pr
{Eh
n(i)
e0= k
}=
λkn
k!e−λn ,
DRAFT
7
where Ehn(i) is the harvested energy by node n in the i-th frame and e0 is the unit of the
arriving energy.
v) The energy arrival rate λn of each node is different from each other. Let Fλ(x) be the
cumulative distribution function (CDF) of λn of the network. Then µλ = Eλ[λn] is the
average energy arrival rate over the network.2
We consider both free space (d2-power loss) and multipath fading (d4-power loss) channel
models. Specifically, we use the free space (fs) model if the distance is shorter than d0 and the
multipath fading (mp) model otherwise. At the RF unit, the amount of energy to transmit one-bit
information to the receiver is
eRF(d) =
ϵfs d ≤ 1,
ϵfsd2 1 < d ≤ d0,
ϵmpd4 d0 < d,
where ϵfs and ϵmp are positive constants and the unit of energy is Joule (J) [11]. Other than this,
an energy eelec for performing signal processing (e.g., the digital coding, modulation, filtering,
and spreading) is needed. Thus, the total energy to transmit a packet from a non-CH node n to
the corresponding CH node m is
ECHn = Lp(eelec + eRF(dmn)). (1)
Let Ehn(i) and Er
n(i) be the harvested energy and the remaining energy of node n at the i-th
frame, respectively. A non-CH node n transmits a packet to its CH node in the assigned slot if
Ern(i) ≥ ECH
n .
Otherwise, the node needs to wait and accumulate more energy. At the end of a frame, the
remaining energy of node n is updated by
Ern(i+ 1) = Er
n(i) + Ehn(i)− ECH
n IErn(i)≥ECH
n, (2)
where IA is the indicator function (1 if A is true and 0 otherwise).
2The scenario where nodes have the same energy arrival rate can be included by setting Fλ(x) = δ(0) as a unit impulse
function.
DRAFT
8
For CH node m, the consumed energy ECHm (i) in a frame consists of 1) the energy to receive
signals from its cluster members, 2) the energy to aggregate the signals, and 3) the energy to
transmit the aggregated packet to the sink. That is,
ECHm (i)=
(Lp
κm−1∑k=1
ITxk(i)(eelec+ eda)+LpedaIκm>1+ en
)ITxm(i), (3)
where κm is the number of nodes in the cluster associated with CH node m, eda is the energy
to aggregate each bit, Txn(i) is the event that node n transmits a packet in frame i, and
en = Lp(eelec + eRF(dn)) (4)
is the required energy for node n to directly transmit a packet to the sink.
After receiving all the packets from its cluster members, the CH node performs some signal
processing operations to aggregate these packets. If the CH node has enough energy, i.e., if
Erm(i) ≥ ECH
m (i), it will forward the aggregated packet to the sink. Otherwise, an outage occurs.
At the end of a frame, the remaining energy of CH node m is updated by
Erm(i+ 1) = Er
m(i) + Ehm(i)− ECH
n (i)IErm(i)≥ECH
n (i).
We also need a measure characterizing the capability of the network in harvesting energy from
ambient environments.
Definition 1: The subsistence ratio of an EH-WSN is the ratio between the sum of the
harvested energy by the network in a frame and the total energy required for every node to
directly transmit a packet to the sink:
ρ =
∑Nn=1 λne0∑Nn=1 en
. (5)
For most of realistic scenarios, ρ is chosen neither too small nor too large in consideration of
operational practicability and cost savings.
III. THE DISTANCE-AND-ENERGY-AWARE ROUTING WITH ENERGY RESERVATION
A. Routing Design Considerations
For a cluster-based routing, most of nodes act as energy-saving non-CH nodes and transmit
packets to their respective CH nodes. However, CH nodes remain energy-consuming since they
need to transmit packets to the remote sink. Thus, it is highly likely that a CH node may suffer
DRAFT
9
from energy shortage. Since the energy shortage event at a CH node blocks the transmission
of the collected information from non-CH members, it degrades the network energy efficiency
significantly. The reduction of energy shortage events at CH nodes is, therefore, crucial for
improving energy efficiency of the network.
The DEARER protocol exploits two key features to address the energy shortage issue of CH
nodes. First, CH nodes are re-assigned periodically according to a head selection rule where
“enabler” nodes, which are in nature resistant to energy shortage, are encouraged to serve as
CH nodes. Second, a portion of the harvested energy during the non-CH mode is reserved for
future use in the CH mode. By doing so, the random energy arrivals are reshaped towards the
desired energy profile that matches with the energy demand.
In the CH selection rule, both the energy harvesting capability and the energy-consuming level
of each node are considered. Specifically, we characterize the relationship between the energy
harvesting capability and the energy-consuming level using two parameters: 1) the average energy
arrival rate λn and 2) the required energy en for a node to transmit a packet to the sink. It is clear
that a node adjacent to the sink (i.e., with smaller en) is more energy efficient in forwarding
packets to the sink. Also, a node with large energy arriving rate λn is preferred to serve as the
energy-consuming CH node. In view of these points, we define the priority of each node serving
as a CH node as
qn =λn
en∑Nk=1
λk
ek
, n = 1, 2, · · · , N. (6)
Moreover, to confront with the randomness of energy arrivals, each node saves a portion psvn
of the harvested energy in the non-CH mode, where the portion psvn is referred to as energy
reservation ratio. The saved energy Esvn is stored into the energy buffer and will be used only
when the node becomes a CH node so that each CH node is more resistant to energy shortage.
B. The Routing Protocol
The overall algorithm is depicted in Table 1. In the network initialization phase, each node
is informed of its slot assignment and priority qn through signaling exchange between nodes
and the sink. In the setup phase of each round, CH nodes are selected and the corresponding
clusters are formed (see Section III-C for details). Next, each CH node measures two types of
energy in the buffer: 1) the remaining energy Ern(NF) corresponding to the residual energy at
DRAFT
10
Algorithm 1 The DEARER protocolInitialization:
1: Sink broadcasts the slots allocation, M , ϵfs, ϵmp, eelec, ϵda at power Ps;
2: Each node decodes the slots allocation, M , ϵfs, ϵmp, eelec, and ϵda;
3: Each node measures the received power to determines dn and en;
4: Each node sends ϑn = λnen
to the sink;
5: The sink calculates and broadcasts∑N
n=1 ϑn;
6: Each node decodes∑N
n=1 ϑn and calculates its own priority qn by (6);
Iteration:
7: for each round l do
8: /∗ Setup Phase: ∗/9: Head selection: using Algorithm 2;
10: Cluster formation: each node joins its nearest CH node;
11: Energy update: each CH nodes updates remaining energy Ern(0) by (7);
12: /∗ Steady Phase: ∗/13: for each frame 1 ≤ i ≤ NF do
14: Each non-CH node transmits a packet if Ern(i) ≥ ECH
n ;
15: Each CH node receives packets from its non-CH members, aggregates them
into one packet, and forwards it to the sink if Erm(i) ≥ ECH
m ;
16: Each node harvests energy and each non-CH nodes reserves energy;
17: Each node updates remaining energy Ern(i) by (2) or (5).
18: end for
19: end for
the end of previous round and 2) the reserved energy Esvn when the node acts as a non-CH node
in previous rounds. Thus, the available energy at the beginning (of the first frame) of a round is
Ern(0) = Er
n(NF) + Esvn . (7)
C. Head Selection
In DEARER, we use the desired number M of CH nodes as a key controlling parameter. To
be specific, the average number of CH nodes in network can be adjusted by varying M .
Let Cn be a flag variable indicating whether node n has recently served as a CH node. Given
Cn, we denote pcondn (l) as the probability of node n being a CH node in the l-th round. At the
beginning of the operation, Cn is initialized as zero. We set Cn = 1 once a node n is elected
as a CH node and reset Cn = 0 after every max{⌈ 1Mqn
− 0.5⌉, 1} rounds. If 1Mqn
≤ 1.5, it is
clear that max{⌈ 1Mqn
− 0.5⌉, 1} = 1, which means that node n elects itself as a CH node with
DRAFT
11
Algorithm 2 The head selection processInitialization:
1: if l mod⌈ 1Mqn
− 0.5⌉ = 0 then
2: Set Cn = 0;
3: end if
Iteration:
4: for each node n do
5: if Cn = 0 then
6: generates a uniformly distributed number x ∼ U(0, 1);
7: if x ≤ pcondn (l) then
8: Elects itself as CH node;
9: Set Cn = 1;
10: Broadcasts its node ID n using a beacon message in its assigned slot;
11: end if
12: end if
13: end for
probability pcondn (l) = 1 in each round. Otherwise, node n serves as a CH node with probability
pcondn (l) =
1
⌈ 1Mqn
− 0.5⌉− l mod⌈ 1Mqn
− 0.5⌉Cn = 0,
0 Cn = 1.
In particular, pcondn (l) is the probability of node n being a CH node on condition that node n
was not a CH node from round⌊
l⌈ 1Mqn
−0.5⌉
⌋⌈1
Mqn− 0.5
⌉+ 1 to round l − 1. Clearly pcond
n (l)
varies from round to round. Moreover, we denote pcondn as the unconditional probability of node
n being a CH node in an arbitrary round.
D. Energy Reservation Policy
As mentioned earlier, in order to prepare more energy for the energy-consuming CH mode,
each node saves a portion of the harvested energy during the non-CH mode. Upon harvesting
a unit of energy, node n immediately puts it into its energy buffer and updates the harvested
energy Ehn(i) (i.e., Eh
n(i) = Ehn(i) + e0). Also, with probability psv
n , the reserved energy Esvn (i)
is updated (i.e., Esvn (i) = Esv
n (i) + e0). At the end of frame i, the harvested energy is reset as
Ehn(i) = Eh
n(i)− Esvn (i).
Note that the saved energy Esvn (i) will not be used until node n is selected as a CH node, at
which time node n measures the total energy of the energy buffer and then updates the remaining
energy Ern(0) (see step 5 in Algorithm 1).
DRAFT
12
It is worth mentioning that the choice of psvn affects the network throughput greatly. To be
specific, if psvn is very small, the node may transmit more packets in the non-CH mode, but this
also results in less energy reservation (and thus higher possibility of energy shortage) in the CH
mode. On the other hand, if psvn is very large, the energy shortage events during the CH mode
are reduced; however, at the same time the transmission of packets in the non-CH mode is also
reduced, which degrades the network throughput. The details on how to properly determine the
energy reservation ratio will be given in Section IV.
IV. PERFORMANCE ANALYSIS
In this section, we analyze the overall efficiency of the DEARER protocol in terms of the
network energy efficiency. We also investigate the transmission efficiency and reliability of each
sensor node in terms of the CH-outage probability and transmission ratio.
A. Definition of Metrics
Our main goal is to investigate how many packets can be successfully delivered to the sink
for the given amount of energy in the network.
Definition 2: The network energy efficiency η is the normalized number of packets that a node
can deliver successfully to the sink per frame per Joule of energy. That is,
η = limt→∞
1
tNF∑N
k=1 λke0
t∑l=1
N∑n=1
NF∑i=1
ITxn(i)ITxm(i), (8)
where node m is the CH node of the cluster to which node n belongs.
We are also interested in how many packets a node actually transmits.
Definition 3: The transmission ratio of a node is the proportion of frames in which the node
has enough energy to transmit a packet:
rn = limt→∞
1
tNF
t∑l=1
NF∑i=1
ITxn(i). (9)
Moreover, the network transmission ratio is the averaged transmission ratio over the network
rnet =1
N
N∑n=1
rn. (9)
Note that the unit of rn and rnet is packets per frame. In essence, the transmission ratio
characterizes how many packets on average a node, either in the non-CH mode or the CH mode,
DRAFT
13
can transmit in a frame, regardless of whether the packets are finally successfully delivered to
the sink. However, if a CH node does not have enough energy to forward the packet to the sink,
then an outage occurs. Thus we also need a metric to measure the outage event.
Definition 4: The CH-outage probability poutn of node n is the probability that the energy of
a node in the CH mode is inadequate to deliver the aggregated packet to the sink. That is,
poutn = lim
t→∞
1
NF∑t
l=1 ICHn(l)
t∑l=1
ICHn(l)
NF∑i=1
(1− ITxn(i)), (10)
where CHn(l) is the event of node n being a CH node. Moreover, the network outage probability
is the averaged CH-outage probability over all nodes in the network:
pout =1
N
N∑n=1
poutn . (11)
In addition, to describe how often a node serves as a CH node, we use the following metric.
Definition 5: The inter-CH time Tn is the waiting time for node n to act again as a CH node.
The metric not only specifies the time for a node to accumulate energy for the upcoming CH
mode, but also will play an important role in our analysis.
B. Inter-CH Time
In this subsection, we present the statistic characterization of inter-CH time. In the proposed
head selection algorithm, a node satisfying 1Mqn
≤ 1.5 is chosen as a CH node in every round.
In this case, the inter-CH time of the node is Tn = 1.
We now focus on the nodes satisfying 1Mqn
> 1.5. Let
τk ∈{k⌈ 1
Mqn− 0.5
⌉+ 1, · · · , (k + 1)
⌈ 1
Mqn− 0.5
⌉}be the round in which node n is a CH node where k =
⌊l
⌈ 1Mqn
−0.5⌉
⌋. Then, according to Definition
5, Tn = τk+1 − τk. In the following, we provide a useful lemma regarding τk and pcondn (see the
definition of pcondn in Section III-C).
Lemma 4.1: Round τk is uniformly distributed in[k⌈ 1
Mqn− 0.5⌉+ 1, (k + 1)⌈ 1
Mqn− 0.5⌉
].
For any round, the unconditional probability of node n being a CH node is
pcondn =
1⌈1
Mqn− 0.5
⌉ . (12)
Proof: See Appendix A.
DRAFT
14
The lemma indicates that each node serves as a CH node once every ⌈ 1Mqn
− 0.5⌉ rounds if1
Mqn≥ 1.5. Using this lemma, we obtain the following proposition.
Proposition 4.2: Let µTn and σ2Tn
be the mean and variance of Tn of node n, respectively.
Then, if 1Mqn
≤ 1.5, node n works as a CH node in every round and µTn = 1. Otherwise,
µTn =⌈ 1
Mqn− 0.5
⌉and σ2
Tn=
⌈1
Mqn− 0.5
⌉2 − 1
6.
Proof: See Appendix B.
It is interesting to compare the proposed head selection method with the pure random head
selection. For the pure random head selection method, each node elects itself as a CH node with
a fixed probability pcondn (see (12)), regardless of whether the node has recently served as a CH
node. It can be shown that
Pr{T rn = k} = (1− pcond
n )k−1pcondn ,
where T rn is the inter-CH time for the pure random head selection method. Hence, the mean and
variance of T rn are given by, respectively,
µT rn=⌈ 1
Mqn− 0.5
⌉and σ2
T rn=⌈ 1
Mqn− 0.5
⌉(⌈ 1
Mqn− 0.5
⌉− 1).
It can be seen that µT rn= µTn and σ2
T rn≫ σ2
Tn, which means that the inter-CH time of the
proposed head selection method suffers less fluctuation compared to that of the pure random
head selection method. Noting that the inter-CH time is essentially the time to accumulate energy
for the use in the next CH mode, the inter-CH time with smaller variance clearly ensures more
stable energy supply to the CH mode, thereby reducing the probability of energy shortage.
C. Energy Consumption of Non-CH Nodes
Let H =∑N
n=1 ICHn be the total number of CH nodes in the network. If we assume that the
average number of CH nodes E[H] is an integer, then we have
E[H] = E
[N∑
n=1
ICHn
]=
N∑n=1
pcondn =
E[H]∑m=1
(κm∑n=1
pcondn
). (13)
From (13), it is clear that for each cluster,∑κm
n=1 pcondn = 1 on average. To simplify the analysis,
we assume that the nodes in the same cluster have a same energy arrival rate µλ. Then the priority
DRAFT
15
of node n being a CH node can be expressed as q′n = µλ
λnqn. Also, the probability of node n
being a CH node is p′n = min{Mq′n, 1}. Thus, 1 =∑κm
n=1 p′n = κmp
′n (i.e., κm = 1
p′n).
Since each node is assumed to be uniformly distributed in the network, the area of cluster m
consists of κm nodes is 4L2κm
Non average. Under the assumption that the cluster has circular
shape, the radius of cluster m is Rm =√
4L2κm
πN. Moreover, since all nodes are uniformly
distributed, the density of a node over the cluster is ρ(x, y) = 1πR2
m. Thus, the average squared
distance d2
nm between a non-CH node and CH node m is
d2
nm=
∫∫ρ(x, y)
√x2 + y2dxdy=
∫ 2π
0
∫ Rm
0
r2rdrdθ
πR2m
=1
2R2
m,
from which we obtain
dnm =
√2L2κm
πN.
By approximating the radius of the cluster to which node n belongs with the radius of the cluster
where node n acts as the CH node, we have κm = κn. Hence, from (1) the average energy that
a non-CH node consumes to transmit a packet to its CH node can be given by
µECHn= Lp(eelec+ eRF(dnm))=Lp
(eelec+ eRF
(√2L2κn
πN
)).
In what follows, we use µECHn
to approximate the energy for node n to transmit a packet to its
CH node.
D. Transmission Ratio
Let rCHn and rCH
n be the transmission ratio of node n in the non-CH mode and the CH mode,
respectively. Then we have
rCHn = lim
n→∞
∑tl=1(1− ICHn)
∑NFi=1 ITxn(i)∑t
l=1(1− ICHn)and rCH
n = limn→∞
∑tl=1 ICHn
∑NFi=1 ITxn(i)∑t
l=1 ICHn
.
Proposition 4.3: Let µTn be the average inter-CH time of node n, κn be the number of nodes
in the cluster associated with CH node n, and poutn be the CH-outage probability defined in (10).
Then, the transmission ratio rCHn of node n in the non-CH mode and the transmission ratio rCH
n
in the CH mode satisfy
rCHn = min
{µTnλne0 − (en + LpedaIκn>1)(1− pout
n )
(µTn − 1)µECHn
+ Lp(κn − 1)(eelec + eda)(1− poutn )
, 1
},
rCHn = 1− pout
n .
(14)
DRAFT
16
Proof: See Appendix C.
In essence, rCHn characterizes the best achievable transmission ratio of node n in the non-CH
mode for a given CH-outage probability poutn . It offers guidelines for determining how much
energy can be used for transmitting packets in the non-CH mode and how much energy should
be saved for the upcoming CH mode (i.e., determining psvn ). To be specific, although rCH
n and
rCHn cannot be maximized simultaneously, the best trade-off can be achieved by choosing for
each node a psvn such that each CH node has exactly the required energy on average to forward
all the information transmitted by its non-CH members. We can also gain useful insights by
investigating the relationship between rCHn , rCH
n , and transmission ratio rn. From Definition 3,
rn = limn→∞
1
tNF
t∑l=1
NF∑i=1
ITxn(i) = limn→∞
1
tNF
(t∑
l=1
(ICHn(l)
+ ICHn(l)
) NF∑i=1
ITxn(i)
)
=
(µTn − 1
µTn
)rCHn +
(1
µTn
)rCHn , (15)
where CHn(l) is the event of node n being a non-CH node and the last equality follows from
pcondn = limn→∞
1t
∑tl=1 ICHn and µTn = 1
pcondn
. Clearly if µTn = 1, node n acts as a CH node in
every round. In this case, the transmission ratio becomes rn = rCHn and rCH
n can be ignored.
E. Energy Reservation Ratio
For node n, the energy reservation ratio psvn can be obtained by investigating its energy
consumption in the non-CH mode.
Proposition 4.4: The energy reservation ratio of node n is
psvn = 1−
µECHnrCHn
λne0. (16)
Proof: Under our energy reservation policy, only a portion of the harvested energy is used
for packet transmission in the non-CH mode while the rest is saved for the use in the next
CH mode. Thus, the energy arrival process, which is a Poisson process with intensity λn, can
be split into two independent subprocesses, more precisely, Poisson processes with intensities
(1−psvn )λn and psv
n λn. This allows us to formulate the energy consumption in the non-CH mode.
Specifically, in the period of Tn − 1 rounds when node n works in the non-CH mode, the total
consumed energy is the difference between the total harvested energy and the saved one, that is,Tn−1∑l=1
NF∑i=1
ITxn(i)ECHn (i) =
Tn−1∑l=1
NF∑i=1
Ehn(i)(1− Isvn(i)). (17)
DRAFT
17
By taking expectation on both sides of (17) and eliminating common terms, we have
µECHnrCHn = (1− psv
n )λne0, (18)
from which we can directly obtain psvn .
Since the consumed energy µECHnrCHn in the non-CH mode is smaller than the totally harvested
energy λne0, psvn is always positive. Moreover, the larger λn is, the more energy a node can save,
and the smaller the corresponding CH-outage will be. In fact, we can uniquely determine the
CH-outage with rCHm and psv
n .
F. The CH-Outage Probability
As mentioned, an outage event occurs at a CH node if it does not have enough energy to
forward the aggregated packet to the sink. The distribution of the available energy at a CH
node is difficult to analyze. Nevertheless, since the available energy at a CH node is the sum of
Poisson random variables, which belong to the exponential family, the distribution can be well
approximated by Gamma distribution, which facilitates proving the following proposition.
Proposition 4.5: The outage probability poutn of node n in the CH mode is
poutn = γ
(kn,
NFµECHn
θn
)− knθnNFµECH
n
γ
(kn + 1,
NFµECHn
θn
). (19)
where γ(k, x) = 1Γ(k)
∫ x
0tk−1e−tdt is the lower incomplete Gamma function and µECH
nis the
average energy that node n consumes per frame in the CH mode,
µECHn
= Lp
((κn − 1)(eelec + eda)r
CHn + edaIκn>1
)+ en, (20)
and kn, θn are given, respectively, by
kn =NFλn(p
svn (µTn − 1) + 1)2
NFλn(psvn )
2σ2Tn
+ psvn (µTn − 1) + 1
and θn = e0 +NFλne0(p
svn )
2σ2Tn
psvn (µTn − 1) + 1
.
Proof: See Appendix D.
For a node n acting as a CH node in a certain round, we can approximate the distribution
of its total available energy E totaln (see (D.24)) using Gamma distribution Γ(kn, θn) (see (D.25)).
By doing so, the average of E totaln can be given as µEtotal
n= knθn. Suppose Z ∼ Γ(kn, 1) is a
Gamma-distributed random variable, we have E[Z] = kn and
γ
(kn,
NFµECHn
θn
)= 1− Pr
{Z >
NFµECHn
θn
}(a)
≥ 1− E[Z]NFµECH
n/θn
= 1−µEtotal
n
NFµECHn
,
DRAFT
18
where (a) uses Markov’s inequality Pr{Z > t} ≤ E[Z]/t [23].
If the network is energy insufficient, i.e., if the subsistence ratios ρ is small, µEtotaln
is much
smaller than the average energy NFµECHn
required for node n to transmit a packet in every frame of
a round when it acts as a CH node, i.e., µEtotaln
≪ NFµECHn
. Thus, we have γ(kn, NFµECHn/θn) ≈ 1
and γ(kn + 1, NFµECHn/θn) ≈ 1. In this case, the CH-outage probability of node n becomes
poutn = 1− knθn
NFµECHn
, (21)
which is relatively simple to interpret.
From Proposition 4.3, 4.4 and 4.5, we can determine rCHn , rCH
n , psvn , and pout
n . Using these results,
we can further compute the network energy efficiency η of the network, which incorporates both
the transmission efficiency of non-CH nodes and the forwarding reliability of CH nodes.
G. Network Energy Efficiency
Recall that the network energy efficiency η is the ratio between the number of successfully
delivered packets in a frame and the total energy harvested by the network in the frame (see
Definition 2). By ergodicity of the energy-arrival process and the law of large numbers, we have
η = limt→∞
1
tNF∑N
k=1 λke0
t∑l=1
N∑n=1
NF∑i=1
ITxn(i)ITxm(i)
=1
Nµλe0
N∑n=1
(µTn − 1
µTn
rCHn (1− pout
m ) +1
µTn
rCHn
)
=1
Nµλe0
N∑n=1
rCHn
µTn
(1 + (µTn − 1)rCHn ),
where in the last equation we have used the approximation poutm = 1− rCH
n .
V. NUMERICAL RESULTS
A. Simulation Setup
We consider N = 1, 000 nodes uniformly distributed over an area of 140 × 140 m2, (i.e.,
L = 70). The sink locates at point (0, 80). The slot length is ts = 0.0016 s, in which a packet
of Lp = 400 bits is transmitted. The steady phase of each round contains NF = 100 frames
and each frame consists of N slots. Following [11], parameters in the energy model are set to
eelec = 50 nJ/bit, eda = 5 nJ/bit/signal, ϵfs = 10 pJ/bit/m2, ϵmp = 0.0013 pJ/bit/m4, and d0 = 87
DRAFT
19
m. As mentioned, we assume that the energy arrival of each node follows the Poisson process.
Without loss of generality, we assume that the energy arrival rate λn of each node is different
from each other and follows Rayleigh distribution with parameter νλ = 1.
B. Direct Transmission and Genie-aided Routing
In direct transmission, each node directly transmits a packet to the sink in the allocated
slot if it has enough energy, in other words, the remaining energy Ern(i) is larger than en =
Lp(eelec + eRF(dn)). At the end of each frame, node n updates the remaining energy by Ern(i+
1) = Ern(i) + Eh
n(i) − enIErn(i)≥en . The transmission ratio of node n is evaluated by rdt
n =
limt→∞1
tNF
∑tl=1
∑NFi=1 ITxn(i), and the network transmission ratio is rdt
net = 1N
∑Nn=1 r
dtn . Note
that there is no outage event in direct transmission since a node will not transmit any packet
unless it has enough energy. However, for comparative purpose, we define pdtout =
1N
∑Nn=1(1−rdt
n )
as the network outage probability of direct transmission.
Similar to DEARER, genie-aided routing is also a cluster-based routing protocol. In the genie-
aided routing, instead of re-assigning CH nodes and establishing clusters once a round, CH
selection and cluster formation is performed in each frame. With the aid of a genie that knows
the real-time remaining energy of each node as well as the distance from each node to the sink,
the CH nodes are selected among the most energy-saving and most energy-sufficient ones. More
precisely, the genie updates the priority for each node being a CH node as
qgen (i) =
Ern(i)en∑N
k=1
Erk(i)
ek
.
Based on the computed priority, the CH nodes are chosen accordingly. Particularly, the number
of CH nodes in the genie-aided routing is chosen to be the number that maximizes the network
energy efficiency (i.e., the number that leads to the most packet delivery to the sink). After the
clusters being formed, non-CH nodes start to transmit packets to their respective CH-node. At
the same time, a CH node will transmit the aggregated packet to the sink if it has the enough
energy to do so.
C. The Calculation of Metrics
It is worth mentioning that parameters poutn , rCH
n , rCHn , and psv
n are related (with each other)
by equations (14), (16), and (19), so that we cannot evaluate them individually. In Table 3, we
DRAFT
20
Algorithm 3 Calculation of poutn , rCH
n , rCHn , and psv
nInitialization:
1: Set maximum iterations K, relative error ε;
2: Set poutn [1] = 01;
Iteration:
3: repeat
4: Solve rCHn [k] and rCH
n [k] from (14), using poutn (k);
5: Solve psvn [k] from (16);
6: Update µECHn
[k] according to (20);
7: Update poutn [k + 1] according to (19);
8: until k = K or |poutn [k]− pout
n [k + 1]| ≤ εpoutn [k + 1]
9: Output: rCHn [K], rCH
n [K], psvn (K), pout
n [K + 1].
present an efficient algorithm to calculate these parameters. The algorithm outputs the solution in
a few iterations. In particular, it terminates either when the maximum iterations K is reached or
when the difference between the output poutn of two adjacent iterations falls below a pre-specified
relative error ε. In our simulations, we set K = 200 and ε = 0.005.
D. Numerical and Simulation Results
In Figure 3(a), we plot network energy efficient η for three protocols under test as a function of
the subsistence ratio ρ. Recall that the desired number M of CH nodes is a controlling parameter
in DEARER. We consider M = 10, 100, and 500 in the simulation. However, since the direct
transmission method is independent of M , its curves of network energy efficiency for different
M coincide with each other. Likewise, since the number of CH nodes in the genie-aided routing
is not controlled by M (in fact, it is chosen to maximize η), the curves of the genie-aided routing
for different M also coincide with each other.
In Figure 3(a), we observe that for all protocols, the network is operated efficiently if ρ ∈
(0.3, 0.4). In particular, DEARER largely outperforms direct transmission in this region. The
gain of DEARER is mainly because 1) the cluster-based routing cuts down long-distance trans-
missions, and 2) the energy reservation policy reschedules the available energy to match with the
energy profile on demand. We also observe that, when ρ is larger than 0.45, the network energy
efficiency of all transmission schemes decreases monotonically with ρ. This is because when
ρ > 0.45, the transmission ratio rnet already approaches one packet/node/frame (see Figure 3(c))
1In Algorithm 3, x[k] is the value of x in the k-th iteration.
DRAFT
21
so that further gain of rnet is marginal irrespective of ρ. Since the network energy efficiency
is normalized by the totally harvested energy, it decreases with ρ. In addition, we observe that
direct transmission is better than DEARER and even performs close to the genie-aided routing
when ρ is very small (i.e., when the network is seriously energy insufficient). This is mainly
due to two reasons. First, if the available energy at each node very little, then for cluster-based
routing protocols, the overhead of packet reception and aggregation is non-negligible. Second,
the advantage of packet aggregation at CH nodes becomes limited because very few packets
could actually be transmitted by non-CH nodes in this case.
Figure 3(b) depicts how the controlling parameter M affects the network energy efficiency.
It can be seen that when the network is not very energy insufficient (e.g., ρ = 0.45 and
ρ = 0.75), DEARER with properly chosen M achieves much higher energy efficiency than
direct transmission, demonstrating advantages of the cluster-based routing and energy reservation
policy. Note that the number of CH nodes in the network increases with M . If M is small, the
number of clusters is small. As a result, the size of each cluster becomes large so that the traffic
at each CH node will be heavy. On the other hand, if M is large, there will be a large number of
CH nodes transmitting packets directly to the sink. In this case, DEARER essentially degrades
to the direct transmission method. Thus, to achieve high network energy efficiency, a moderate
M is preferred. However, when the network is seriously energy insufficient, (e.g., ρ = 0.15),
DEARER has lower network energy efficiency when compared with the direct transmission
method, as shown in Figure 3(a). Nevertheless, the improvement in energy efficiency can be
achieved by choosing very large M . In doing so, the network energy efficiency of DEARER
approaches that of direct transmission.
We present more details on the network transmission ratio (rnet =1N
∑Nn=1 rn) and network
outage probability (pout =1N
∑Nn=1 p
outn ) in Figure 3(c) and 3(d), respectively. As is clear from
the figure, DEARER has much higher transmission ratio and also much smaller CH-outage
probability than direct transmission when ρ = 0.45 and ρ = 0.75. This means that under
the DEARER protocol, the sensor nodes transmit more packets, and the packet forwarding at
CH nodes is also more reliable. When ρ = 0.15, while the transmission ratio of DEARER is
higher than that of direct transmission (see Figure 3(c)), its CH-outage probability is larger (see
Figure 3(d)), which results in low network energy efficiency (see Figure 3(a)).
Figure 4 displays the energy reservation ratio psvn of each node. The psv
n of nodes are sorted in
DRAFT
22
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Subsistence ratio ρ
Netw
ork
energ
y e
ffic
iency η
M = 10
M = 100
M = 500
x 104
(a) Network energy efficiency as a function of ρ (M = 100)
101
102
103
0.2
0.4
0.6
0.8
1
1.2
1.4
Desired number M of CH nodes
Netw
ork
energ
y e
ffic
iency η
ρ = 0.15
ρ = 0.45
ρ = 0.75
x 104
(b) Network energy efficiency as a function of M (xs = 80)
101
102
103
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Desired number M of CH nodes
Netw
ork
tra
nsm
issio
n r
atio r
net
ρ = 0.15
ρ = 0.45
ρ = 0.75
(c) Network transmission ratio as a function of M (xs = 80)
101
102
103
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Desired number M of CH nodes
Netw
ork
outa
ge p
robabili
ty p
out
ρ = 0.15
ρ = 0.45
ρ = 0.75
(d) Network outage probability as a function of M (xs = 80)
Figure 3. Performance comparison between direct transmission, genie-aided routing, and the proposed DEARER protocol,
where the curves marked by o, ◃, +, and � correspond to DEARER in theory, DEARER by Monte Carlo, direct transmission,
and genie-aided routing, respectively.
an ascending order. When the subsistence ratio ρ is very small, the available energy is limited
for most of nodes so that there is little energy to save. Thus, the energy reservation ratio of
most nodes is zero. It is seen that when the energy subsistence ratio is increased from ρ = 0.15
to ρ = 0.45 (or from ρ = 0.45 to ρ = 0.75), psvn for most of nodes becomes larger. Moreover,
if M is increased from M = 10 to M = 100, psvn also becomes larger. This is due to the fact
that when M increases, each node serves as a CH node more frequently and hence needs more
DRAFT
23
0 100 200 300 400 500 600 700 800 900 10000
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Serial number of node
Reserv
ation r
atio o
f nodes p
nsv
M = 10, ρ = 0.15
M = 10, ρ = 0.45
M = 100, ρ = 0.45
M = 100, ρ = 0.75
M = 500, ρ = 0.75
Figure 4. The energy reservation ratio of DEARER where xs = 80 m.
energy for the CH mode. When ρ = 0.75 and we increase M = 100 to M = 500, however,
the psvn becomes smaller and even zero. This is because in these cases most of nodes act as CH
nodes every round and hence do not save energy.
VI. CONCLUSION
In this paper, we have proposed a routing protocol called DEARER for EH-WSNs. By encour-
aging enabler nodes to serve more frequently as CH nodes and to save a portion of the harvested
energy for the CH mode, DEARER outperforms direct transmission and approaches the genie-
aided routing for most of the realistic scenarios. We have established a mathematically tractable
model and presented a detailed analysis of DEARER by employing statistical approximations.
These approximations are reasonable as the underlying quantities converge to their respective
averages when the operation time goes to infinity.
In DEARER, the desired number M of CH nodes can be tuned to improve the network energy
efficiency. From numerical and simulation results, we have shown that when the network is not
very energy insufficient (i.e., the substance ratio ρ is not very small), with properly chosen M ,
DEARER largely outperform direct transmission. Whereas, when the network is seriously energy
insufficient, the direct transmission method performs better. In fact, direct transmission performs
as well as the genie-aided routing in this case.
We have also investigated how much energy a node should save in its non-CH mode in
terms of the energy reservation ratio. To make full use of the harvested energy, we choose
DRAFT
24
the energy reservation ratio such that in an average sense, each non-CH node reaches its best
achievable transmission ratio while each CH node has the exact amount of energy to transmit
the aggregated packet to the sink. We would like to point out that in the DEARER protocol,
the energy reservation ratio is constant over time. In many application scenarios, however, real-
time energy status is accessible (e.g., through message exchanges). By utilizing the real-time
energy status, a dynamic energy saving (or an online energy scheduling) may lead to further
improvement of the network energy efficiency. We would also like to point out that our model
and analysis of the DEARER protocol are built upon approximations in the average sense and
do not incorporate the game-playing among nodes. Note that selfish nodes may achieve their
optimal individual transmission ratio if they evade serving as CH nodes so that the transmission
scheme would degrade to the direct transmission. We believe that a game-theoretic formulation
of the cluster-based routing may bring new insights into the design of routing algorithms in
EH-WSNs. Our future research will be directed towards these avenues.
APPENDIX A
PROOF OF LEMMA 4.1
Proof: For simplicity, we consider the head selections from round 1 to round ⌈ 1Mqn
− 0.5⌉.
Let Al be the event that node n is a CH node in round l and Al be its complementary event. It
is seen that pcondn (l) = 1
⌈ 1Mqn
−0.5⌉−lis actually the probability of the event {Al|A1, A2, · · · , Al−1}.
Therefore, the probability distribution of τ1 can be given by
Pr{τ1 = l} = Pr{Al, A1, · · · , Al−1} = Pr{A1}Pr{A2|A1} · · ·Pr{Al|A1, · · · ,Al−1}
=
(1− 1
⌈ 1Mqn
− 0.5⌉ − 1
)(1− 1
⌈ 1Mqn
− 0.5⌉ − 2
)· · ·(
1− 1
⌈ 1Mqn
− 0.5⌉ − (l − 1)
)1
⌈ 1Mqn
− 0.5⌉ − l
=1
⌈ 1Mqn
− 0.5⌉,
which implies that τ1 follows the discrete uniform distribution. Likewise, one can also prove
that τk is a discrete uniformly distributed random variable.
Noting that Pr{Al} = Pr{Al, A1, · · · , Al−1} + Pr{Al, A1, · · · , Al−1} during the period of
DRAFT
25
round 1 to round ⌈ 1Mqn
− 0.5⌉, we have
pcondn = Pr{Al} = Pr{Al, A1, · · · , Al−1}+ Pr{Al, A1, · · · , Al−1} =
1
⌈ 1Mqn
− 0.5⌉, (A.22)
where Pr{Al, A1, A2, · · · , Al−1} is zero because node n cannot serve as a CH node more than
once in this period. In addition, the head selection for each node n during the period of round
k⌈ 1Mqn
−0.5⌉+1 to round (k+1)⌈ 1Mqn
−0.5⌉ is independent of the head selections before round
k⌈ 1Mqn
− 0.5⌉+ 1. Thus, we can conclude that (A.22) holds for any k ≥ 0.
APPENDIX B
PROOF OF PROPOSITION 4.2
Proof: For simplicity, we consider the head selections during the period of round 1 to round
⌈ 1Mqn
− 0.5⌉ and the inter-CH time Tn = τ2 − τ1. From Lemma 4.1, τ1 and τ2 are variables
uniformly distributed in[1, ⌈ 1
Mqn− 0.5⌉
]and
[⌈ 1Mqn
− 0.5⌉ + 1, 2⌈ 1Mqn
− 0.5⌉], respectively.
Denoting τ ′1 = ⌈ 1Mqn
− 0.5⌉ − τ1 and τ ′2 = τ2 − ⌈ 1Mqn
− 0.5⌉, we have Tn = τ ′1 + τ ′2, and
Pr{τ ′1 ≤ j} = Pr
{⌈ 1
Mqn− 0.5⌉ − τ1 ≤ j
}= Pr
{τ1 ≥ ⌈ 1
Mqn− 0.5⌉ − j
}= 1−
⌈ 1Mqn
− 0.5⌉ − j − 1
⌈ 1Mqn
− 0.5⌉
=j + 1
⌈ 1Mqn
− 0.5⌉,
which means that τ ′1 is discrete and uniformly distributed in[0, ⌈ 1
Mqn− 0.5⌉ − 1
]. Since τ1 is
independent of τ2, τ ′1 is also independent of τ ′2. Note that the mean and variance of a discrete
uniformly distributed random variable X ∼ U(a, b) is E[X] = a+b2
and D[X] = (b−a+1)2−112
. Thus,
E[Tn] = E[τ ′1] + E[τ ′2] =⌈ 1
Mqn− 0.5
⌉and D[Tn] = D[τ ′1] + D[τ ′2] =
⌈ 1Mqn
− 0.5⌉2 − 1
6,
which completes the proof.
APPENDIX C
PROOF OF PROPOSITION 4.3
Proof: For an arbitrarily chosen period of Tn rounds, node n serves as a non-CH node for
Tn − 1 rounds and as a CH node for one round. Using (10) together with ergodicity of energy
harvesting process and the law of large numbers, we have rCHn = 1− pout
n .
DRAFT
26
In an energy reservation policy that makes the best use of the harvested energy (without any
waste or shortage of energy), the totally consumed energy of nodes in both the non-CH and CH
modes equals to the totally harvested energy. That is,Tn−1∑l=1
NF∑i=1
ITxn(i)ECHn (i) +
NF∑i=1
ITxn(i)ECHn (i) =
Tn∑l=1
NF∑i=1
Ehn(i). (C.23)
By taking expectation on both sides of (C.23), we have
(µTn − 1)rCHn ECH
n (i) + ECHn (i)rCH
n = µTnEhn(i).
Since the transmission ratio of a node depends largely on its distance to the sink, the nodes
within the same cluster have the same transmission ratio on average. Thus, using the approxi-
mation ECHn (i) ≈ µECH
n, we have(
Lp(κn − 1)(eelec + eda)rCHn + LpedaIκn>1 + en
)rCHn + (µTn − 1)µECH
n)rCH
n = µTnλne0,
where κn is the number of nodes in the cluster associated with the CH node n. By rearranging
terms, we obtain the first term in the parenthesis of (14).
Since a node can transmit at most one packet in a frame under the DEARER protocol,
the transmission ratio is upper bounded by one, irrespective of the energy arrival rate, which
establishes the proposition.
APPENDIX D
PROOF OF PROPOSITION 4.5
Proof: By ergodicity of the energy arrival process, head selection process, and energy saving
process, we have
poutn =
NF−1∑j=0
(NF − j
NFPr
{j µECH
n< Er
n(0) +
NF∑i=1
Ehn(i) < (j + 1)µECH
n
}),
where µECHn
= E[ECHn ] is the average energy consumption of CH node n in a frame.
To prove Proposition 4.5, we first need to find the probability distribution of the energy
available for a CH node. The Gamma distribution and the moment matching method are presented
in the following.
DRAFT
27
1) Gamma distribution and moment matching: We denote the Gamma distributed random
variable with shape k > 0 and θ > 0 by X ∼ Γ(k, θ). The probability density function (pdf)
using the shape-scale parametrization is [20]
fX(x) = xk−1 c−x/θ
θkΓ(k),
where Γ(k) is the gamma function evaluated at k. The first and second moments and variance are
E[X] = kθ, E[X2] = k(k + 1)θ2, and D[X] = kθ2.
It is well known that a distribution with µ = E[Y ], µ(2) = E[Y 2] and variance σ2 = µ(2) − µ2
can be approximated with a Gamma distribution Γ(k, θ) by matching the first and second order
moments where k and θ are given, respectively, by
k =µ2
σ2and θ =
σ2
µ.
This method is known as moment matching [20].
2) Probability distribution of the energy available at CH nodes: When node n is selected as
a CH node, the total amount of available energy for the operation in the CH mode is given by
E totaln = Er
n(0) + Ehn(1 : NF), (D.24)
where Ern(0) =
∑Tn−1l=1
∑NFi=1E
svn (i) is the saved energy when node n is in the non-CH mode and
Ehn(1 : NF) =
∑NFi=1E
hn(i) is the harvested energy in the CH mode. In particular, {Esv
n (i), i =
1, 2, · · ·NF} for each round is a Poisson process with parameter psvn λn.
Note that if X is a Poisson distributed random variable with parameter λ, then for any constant
c = 0, the moment generating function (MGF) of cX is GcX(s) = E[escX
]= exp (λ(ecs − 1)).
Thus, the MGF of Esvn (i) is GEsv
n(s) = exp (psv
n λ(ee0s − 1)) and the MGF of Er
n(0) is
GErn(0)(s) = E
[exp
(sTn−1∑l=1
NF∑i=1
Esvn (i)
)]= ETn
[Tn−1∏l=1
NF∏i=1
EEsvn
[esE
svn (i)]]
= ETn [exp ((Tn − 1)NFpsvn λn(e
e0s − 1))]
= exp(φ(s))GTn(φ(s)),
where φ(s) = NFpsvn λn(e
e0s − 1) and GTn(s) is the MGF of inter-CH time Tn. Since Tn is the
sum of two independent uniformly distributed random variables (Tn = τ ′1 + τ ′2), we have
GsTn(s) = et
(et(⌈
1Mqn
−0.5⌉−1) − 1
t(⌈ 1Mqn
− 0.5⌉ − 1)
)2
.
DRAFT
28
Given the MGF of a random variable X , the moments can be obtained as E[x(k)] = G(k)X (0)(s)|s=0.
Thus, the first and second moments of Ern(0) can be given by
E[Ern(0)] = cn(µTn − 1) and E[(Er
n(0))2] = c2n(µ
(2)Tn
− 2µTn + 1) + cne0(µTn − 1),
respectively, where cn = NFpsvn λne0, µTn = (Gr
Tn(s))′|s=0 = E[Tn], and µ
(2)Tn
= (GrTn(s))′′|s=0 =
E[T 2n ]. This implies that the variance of Er
n(0) is
D[Ern(0)] = c2nD[Tn] + cne0(µTn − 1).
Likewise, the MGF of Ehn(1 : NF) is
GEhn(1:NF)(s) = E
[esE
hn(1:NF)
]= exp (NFλn(e
e0s − 1)),
and its mean and variance are
E[Ehn(1 : NF)] = NFλne0 and D[Eh
n(1 : NF)] = NFλne20.
Since Ern(0) and Eh
n(1:NF) are independent of each other, the mean and variance of E totaln are
µEtotaln
= E[Ern(0)] + E[Eh
n(1 : NF)] and σ2Etotal
nD[Er
n(0)] + D[Ehn(1 : NF)].
According to the moment matching method, the distribution of E totaln can be approximated by
a Gamma distribution Γ(kn, θn) with the same moments where
kn =µ2Etotal
n
σ2Etotal
n
and θn =σ2Etotal
n
µEtotaln
. (D.25)
3) CH-outage probability: By ergodicity of the energy arrival process, the head selection
process, and the energy saving process, one can rewritten the CH-outage probability in (10) as
poutn = lim
t→∞
1
NF∑t
l=1 ICHn(l)
t∑l=1
ICHn(l)
NF∑i=1
(1− ITxn(i))
= limt→∞
t∑tl=1 ICHn(l)
limt→∞
1
tNF
t∑l=1
ICHn(l)
NF∑i=1
(1−ITxn(i))
(a)=
µTnpcondn
NF
NF∑i=1
ETn,Ehn(i),E
svn (i)
[I{Er
n(i)<ECHn (i)}
](b)=
NF−1∑j=0
NF − j
NFEEtotal
n
[Pr{jµECH
n<E total
n < (j + 1)µECHn
}]=
NF−1∑j=0
NF − j
NFθknn Γ(kn)
∫ (j+1)µECHn
j µECHn
xkn−1e−xθn dx
DRAFT
29
where (a) follows the weak law of large numbers [23] and in (b), we have used ECHn = µECH
nto
approximate the energy consumed by CH node n in a frame.
By rearranging the items, we have
poutn =
1
θknn Γ(kn)
∫ NF µECHn
0
xkn−1e−xθn dx−
NF−1∑j=0
∫ (j+1)µECHn
jµECHn
jµECHnxkn−1e−
xθn dx
NFθknn Γ(kn)µECHn
≥ γ
(kn,
NFµECHn
θn
)− knθn
NFµECHn
γ
(kn + 1,
NFµECHn
θn
), (D.26)
where γ(k, x) = 1Γ(k)
∫ x
0tk−1e−tdt is the lower incomplete Gamma function.
On the other hand,
poutn =
NF + 1
NFθknn Γ(kn)
∫ NF µECHn
0
xkn−1e−xθn dx−
NF+1∑j=0
∫ (j+1)µECHn
j µECHn
(j + 1)µECHnxkn−1e−
xθn dx
NFθknn Γ(kn)µECHn
≤ NF + 1
NFγ
(kn,
NFµECHn
θn
)− knθn
NFµECHn
γ
(kn + 1,
NFµECHn
θn
). (D.27)
Using (D.26) and (D.27), and also noting that NF is generally large, we have
poutn = γ
(kn,
NFµECHn
θn
)− knθn
NFµECHn
γ
(kn + 1,
NFµECHn
θn
).
This completes the proof.
REFERENCES
[1] J. K. Hart and K. Martinez, “Environmental Sensor Networks: A revolution in the earth system science?” Earth-Science
Reviews, vol. 78, no. 3, pp. 177–191, Oct. 2006.
[2] P. E. Ross, “Managing care through the air,” IEEE Spectrum, vol. 41, no. 12, pp. 26–31, Dec. 2004.
[3] K. K. Khedo, R. Perseedoss, and A. Mungur, “A wireless sensor network air pollution monitoring system,” Int. J. of
Wireless and Mobile Netw., vol. 2, no. 2, pp. 31–45, May 2010.
[4] L. Mainetti, L. Patrono, and A. Vilei, “Evolution of wireless sensor networks towards the Internet of Things: A survey,”
in Proc. 19th Int. Conf. on Software, Telecommun. and Comput. Netw. (SoftCOM’11), Split, Croatia, Sep. 2011, pp. 1–6.
[5] A. Kouche, “Towards a wireless sensor network platform for the internet of things: Sprouts WSN platform,” in Proc. IEEE
Int. Conf. on Commun. (ICC12), Ottawa, ON, Canada, Jun. 2012, pp. 632–636.
[6] S. Sudevalayam and P. Kulkarni, “Energy harvesting sensor nodes: survey and implications,” IEEE Commun. Surveys Tuts.,
vol. 13, no. 3, pp. 443–461, Mar. 2011.
[7] Z. Niu, X. Guo, S. Zhou, and P. R. Kumar, “Characterizing energy-delay tradeoff in hyper-cellular networks with base
station sleeping control,” IEEE J. Sel. Areas Commun., vol. 33, no. 4, pp. 641–649, Apr. 2015.
[8] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Trans.
Wireless Commun., vol. 12, no. 5, pp. 1989–2001, May 2013.
DRAFT
30
[9] C. Intanagonwiwat, R. Govindan, and D. Estrin, “Directed diffusion: a scalable and robust communication paradigm for
sensor networks,” in Proc. ACM Mobi-Com 2000, Boston, MA, Aug. 2000, pp. 56–67.
[10] D. Braginsky and D. Estrin, “Rumor routing algorithm for sensor networks,” in Proc. 1st Wksp. Sensor Netw. and Apps.,
Atlanta, GA, Oct. 2002, pp. 22–31.
[11] W. B. Heinzelman, A. P. Chandrakasan, and H. Balakrishnan, “An application-specific protocol architecture for wireless
microsensor networks,” IEEE Trans. Wireless Commun., vol. 1, no. 4, pp. 660–670, Oct. 2002.
[12] A. Manjeshwar and D. P. Agarwal, “TEEN: a routing protocol for enhanced efficiency in wireless sensor networks,” in
Proc. 1stInt. Wksp. on Paral. and Distrib. Comput. Issues in Wireless Netw. and Mobile Comput., San Francisco, CA, Apr.
2001.
[13] F. Kuhn, R. Wattenhofer, and A. Zollinger, “Worst-case optimal and average-case efficient geometric Ad hoc routing, in
Proc. 4th ACM Intl. Conf. Mobile Comput. and Netw., New York, NY, USA, Jun. 2003, pp. 267–78.
[14] J. Li and D. Liu, “DPSO-based clustering routing algorithm for energy harvesting wireless sensor networks,” in Proc. Int.
Conf. on Wireless Commun. & Signal Process. (WCSP’15), Nanjing, China, Oct. 2015, pp. 1–5.
[15] Z. A. Eu, H.-P. Tan and W. K. G. Seah, “Opportunistic routing in wireless sensor networks powered by ambient energy
harvesting,” Comput. Netw., vol. 54, no. 17, pp. 2943–2966, Dec. 2010.
[16] Z. A. Eu and H.-P Tan, “Adaptive opportunistic routing protocol for energy harvesting wireless sensor networks” in Proc.
IEEE Int. Conf. on Commun. (ICC12), Ottawa, ON, Canada, Jun. 2012, pp. 318–322.
[17] E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Inform. Theory, vol. 51, no. 12, pp. 4203–4215,
Dec. 2005.
[18] D. L. Donoho, M. Elad, and V. N. Temlyakov, “Stable recovery of sparse overcomplete representations in the presence of
noise,” IEEE Trans. Inform. Theory, vol. 52, no. 1, pp. 6–18, Jan. 2006.
[19] M. K. Jakobsen, J. Madsen, and M. R. Hansen. “DEHAR: a distributed energy harvesting aware routing algorithm for
Ad-hoc multi-hop wireless sensor networks,” in Proc. IEEE Int. Symp. World of Wireless Mobile and Multimedia Netw.
(WoWMoM’10), Montreal, QC, Canada, Jun. 2010, pp. 1–9.
[20] R. W. Heath, M. Kountouris, and T. Bai, “Modeling heterogeneous network interference using poisson point processes,”
IEEE Trans. Signal Process., vol. 61, no. 16, pp. 4114–4125, Aug. 2013.
[21] Wikipedia contributors, “Button cell,” Wikipedia, The Free Encyclopedia, https://en.wiki pedia.org/w/
index.php?title=Button cell&oldid=689713099.
[22] J. M. Jornet and I. F. Akyildiz, “Joint energy harvesting and communication analysis for perpetual wireless nanosensor
networks in the terahertz band,” IEEE Trans. Nanotech., vol. 11, no. 3, pp. 570–580, May 2012.
[23] G. Grimmet and D. Stirzaker, Probability and Random Processes, Oxford University Press, 2001.
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